We briefly recollect some definitions in [[1]}, Section 2 and Appendix A] as follows. Let \(I\mathcal {Z}\)  be the inertia stack and \(\pi _{0}:I\mathcal {Z}\rightarrow \mathcal {Z}\)  be the natural projection. We have a decomposition of \(I\mathcal {Z}\)  as a disjoint union \(I\mathcal {Z}=\coprod _{i\in \mathcal {I}}\mathcal {Z}_{i}\)  for some index set \(\mathcal {I}\) , and a canonical involution \(\iota : I\mathcal {Z}\rightarrow I\mathcal {Z}\) . Given a vector bundle \(W\)  on \(I\mathcal {Z}\) , there exists a decomposition \(W=\bigoplus _{\vartheta }W^{(\vartheta )}\)  where \(W^{(\vartheta )}\)  is an eigenbundle with the eigenvalue \(\vartheta \) . And we have a map \(\hat{\rho }: K(I\mathcal {Z})\rightarrow K(I\mathcal {Z})_{\mathbb {C}}\)  defined by \(\hat{\rho }:=\sum _{\vartheta }\vartheta W^{(\vartheta )}\) . Define \(\widetilde{\mathrm {ch}}: K(\mathcal {Z})\rightarrow A^*(I\mathcal {Z})_{\mathbb {C}}\)  by \(\widetilde{\mathrm {ch}}(V):=\mathrm {ch}(\hat{\rho }(\pi _{0}^*V))\)  where \(\mathrm {ch}\)  is the usual Chern character. By [[2]}, Section 7.3], we have the orbifold (Chen-Ruan) cohomology of \(\mathcal {Z}\)  as \(A^*_{\mathrm {orb}}(\mathcal {Z}):=\bigoplus _{i} A^{*-\mathrm {age}_{i}}(\mathcal {Z}_{i})\)  where \(\mathrm {age}_{i}\)  is the degree shift number. The orbifold Chern character \(\widetilde{\mathrm {ch}}^{\mathrm {orb}}: K(\mathcal {Z})\rightarrow A^*_{\mathrm {orb}}(\mathcal {Z})\)  is defined by \(\widetilde{\mathrm {ch}}_{k}^{\mathrm {orb}}|_{\mathcal {Z}_{i}}:=\widetilde{\mathrm {ch}}_{k-\mathrm {age}_{i}}|_{\mathcal {Z}_{i}}\) .
Since \(\mathrm {PT}_{\mathcal {Z}/\mathbb {C}}^P\)  is a fine moduli space (see [[3]}, Theorem 5.20]), there is a universal complex \(\hat{\mathbb {I}}^\bullet :=\lbrace \mathcal {O}_{\mathcal {Z}\times \mathrm {PT}_{\mathcal {Z}/\mathbb {C}}^P}\rightarrow \mathbb {F}\rbrace \)  where \(\mathbb {F}\)  is the universal sheaf on \(\mathcal {Z}\times \mathrm {PT}_{\mathcal {Z}/\mathbb {C}}^P\) . Since \(\mathbb {F}\)  has a locally free resolution of finite length by [[3]}, Lemma 5.9], then \(\widetilde{\mathrm {ch}}^{\mathrm {orb}}(\mathbb {F})\in A^*(I\mathcal {Z}\times \mathrm {PT}_{\mathcal {Z}/\mathbb {C}}^P)\) .
For any \(\gamma \in A^l_{\mathrm {orb}}(\mathcal {Z})\) , define the operator \(\widetilde{\mathrm {ch}}_{k+2}^{\mathrm {orb}}(\gamma ): A_{*}(\mathrm {PT}_{\mathcal {Z}/\mathbb {C}}^P)\rightarrow A_{*-k+1-l}(\mathrm {PT}_{\mathcal {Z}/\mathbb {C}}^P)\)  to be
\(\widetilde{\mathrm {ch}}_{k+2}^{\mathrm {orb}}(\gamma )(\xi ):=\pi _{2*}\left(\widetilde{\mathrm {ch}}^{\mathrm {orb}}_{k+2}(\mathbb {F})\cdot \iota ^*\pi _{1}^*\gamma \cap \pi _{2}^*\xi \right)\) 
